Elongated triangular bipyramid

Short description: 14th Johnson solid; triangular prism capped with tetrahedra

Elongated triangular bipyramid

Type| Johnson
J13 – J14 – J15
Faces| 6 triangles
3 squares
Edges| 15
Vertices| 8
Vertex configuration| 2(33)
6(32.42)
Symmetry group| D3h, [3,2], (*322)
Rotation group| D3, [3,2]+, (322)
Dual polyhedron| Triangular bifrustum
Properties| convex
Net

In geometry, the elongated triangular bipyramid (or dipyramid) or triakis triangular prism a polyhedron constructed from a triangular prism by attaching two tetrahedrons to its bases. It is an example of Johnson solid.

## Contents

* 1 Construction
* 2 Properties
* 3 Appearances
* 4 References
* 5 External links

## Construction

The elongated triangular bipyramid is constructed from a triangular prism by attaching two tetrahedrons onto its bases, a process known as the elongation.[1] These tetrahedrons cover the triangular faces so that the resulting polyhedron has nine faces (six of them are equilateral triangles and three of them are squares), fifteen edges, and eight vertices.[2] A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The elongated bipyramid is one of them, enumerated as the fourteenth Johnson solid J14.[3]

## Properties

File:3D Johnson J14.stl3D model of an elongated triangular bipyramid

The surface area of an elongated triangular bipyramid A is the sum of all polygonal face's area: six equilateral triangles and three squares. The volume of an elongated triangular bipyramid V can be ascertained by slicing it off into two tetrahedrons and a regular triangular prism and then adding their volume. The height of an elongated triangular bipyramid h is the sum of two tetrahedrons and a regular triangular prism' height. Therefore, given the edge length a, its surface area and volume is formulated as:[2][4] A=(332+3)a2≈5.598a2,V=(26+32)a3≈0.669a3,h=(263+1)⋅a≈⋅2.633a.

It has the same three-dimensional symmetry group as the triangular prism, the dihedral group D3h of order twelve. The dihedral angle of an elongated triangular bipyramid can be calculated by adding the angle of the tetrahedron and the triangular prism:[5]

* the dihedral angle of a tetrahedron between two adjacent triangular faces is arccos⁡(13)≈70.5∘;
* the dihedral angle of the triangular prism between the square to its bases is π2=90∘, and the dihedral angle between square-to-triangle, on the edge where tetrahedron and triangular prism are attached, is arccos⁡(13)+π2≈160.5∘;
* the dihedral angle of the triangular prism between two adjacent square faces is the internal angle of an equilateral triangle π3=60∘.

## Appearances

The nirrosula, an African musical instrument woven out of strips of plant leaves, is made in the form of a series of elongated bipyramids with non-equilateral triangles as the faces of their end caps.[6]

## References

1. ↑ Rajwade, A. R. (2001), Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem, Texts and Readings in Mathematics, Hindustan Book Agency, p. 84-89, doi:10.1007/978-93-86279-06-4, ISBN 978-93-86279-06-4, https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84 .
2. ↑ 2.0 2.1 Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8 .
3. ↑ Uehara, Ryuhei (2020), Introduction to Computational Origami: The World of New Computational Geometry, Springer, p. 62, doi:10.1007/978-981-15-4470-5, ISBN 978-981-15-4470-5, https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62 .
4. ↑ Sapiña, R., "Area and volume of the Johnson solid J8" (in es), Problemas y Ecuaciones, ISSN 2659-9899, https://www.problemasyecuaciones.com/geometria3D/volumen/Johnson/J8/calculadora-area-volumen-formulas.html, retrieved 2020-09-09 .
5. ↑ "Convex polyhedra with regular faces", Canadian Journal of Mathematics 18: 169–200, 1966, doi:10.4153/cjm-1966-021-8 .
6. ↑ Gerdes, Paulus (2009), "Exploration of technologies, emerging from African cultural practices, in mathematics (teacher) education", ZDM Mathematics Education 42 (1): 11–17, doi:10.1007/s11858-009-0208-2 .

## External links

* Eric W. Weisstein, Elongated triangular bipyramid (Johnson solid) at MathWorld.

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Johnson solids

Pyramids, cupolae and rotundae|

* square pyramid
* pentagonal pyramid
* triangular cupola
* square cupola
* pentagonal cupola
* pentagonal rotunda

Modified pyramids|

* elongated triangular pyramid
* elongated square pyramid
* elongated pentagonal pyramid
* gyroelongated square pyramid
* gyroelongated pentagonal pyramid
* triangular bipyramid
* square bipyramid
* pentagonal bipyramid
* elongated triangular bipyramid
* elongated square bipyramid
* elongated pentagonal bipyramid
* gyroelongated square bipyramid

Modified cupolae and rotundae|

* elongated triangular cupola
* elongated square cupola
* elongated pentagonal cupola
* elongated pentagonal rotunda
* gyroelongated triangular cupola
* gyroelongated square cupola
* gyroelongated pentagonal cupola
* gyroelongated pentagonal rotunda
* gyrobifastigium
* triangular orthobicupola
* square orthobicupola
* square gyrobicupola
* pentagonal orthobicupola
* pentagonal gyrobicupola
* pentagonal orthocupolarotunda
* pentagonal gyrocupolarotunda
* pentagonal orthobirotunda
* elongated triangular orthobicupola
* elongated triangular gyrobicupola
* elongated square gyrobicupola
* elongated pentagonal orthobicupola
* elongated pentagonal gyrobicupola
* elongated pentagonal orthocupolarotunda
* elongated pentagonal gyrocupolarotunda
* elongated pentagonal orthobirotunda
* elongated pentagonal gyrobirotunda
* gyroelongated triangular bicupola
* gyroelongated square bicupola
* gyroelongated pentagonal bicupola
* gyroelongated pentagonal cupolarotunda
* gyroelongated pentagonal birotunda

Augmented prisms|

* augmented triangular prism
* biaugmented triangular prism
* triaugmented triangular prism
* augmented pentagonal prism
* biaugmented pentagonal prism
* augmented hexagonal prism
* parabiaugmented hexagonal prism
* metabiaugmented hexagonal prism
* triaugmented hexagonal prism

Modified Platonic solids|

* augmented dodecahedron
* parabiaugmented dodecahedron
* metabiaugmented dodecahedron
* triaugmented dodecahedron
* metabidiminished icosahedron
* tridiminished icosahedron
* augmented tridiminished icosahedron

Modified Archimedean solids|

* augmented truncated tetrahedron
* augmented truncated cube
* biaugmented truncated cube
* augmented truncated dodecahedron
* parabiaugmented truncated dodecahedron
* metabiaugmented truncated dodecahedron
* triaugmented truncated dodecahedron
* gyrate rhombicosidodecahedron
* parabigyrate rhombicosidodecahedron
* metabigyrate rhombicosidodecahedron
* trigyrate rhombicosidodecahedron
* diminished rhombicosidodecahedron
* paragyrate diminished rhombicosidodecahedron
* metagyrate diminished rhombicosidodecahedron
* bigyrate diminished rhombicosidodecahedron
* parabidiminished rhombicosidodecahedron
* metabidiminished rhombicosidodecahedron
* gyrate bidiminished rhombicosidodecahedron
* tridiminished rhombicosidodecahedron

Elementary solids|

* snub disphenoid
* snub square antiprism
* sphenocorona
* augmented sphenocorona
* sphenomegacorona
* hebesphenomegacorona
* disphenocingulum
* bilunabirotunda
* triangular hebesphenorotunda

(See also List of Johnson solids, a sortable table)

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